The US does not need Sweden's involvement to extradite and try Assange. The rape charges are separate and irrelevant to the charges he faces in the US, and whether the UK would even grant his extradition to the US is not a sure thing.

These two cases are unrelated, and the only reason they have been conflated is a concerted effort on the part of Assange supporters to discredit the accusations of sexual misconduct by lumping them in with his WikiLeaks activities and make up a whole lot of guff about CIA plots rather than just accept that he should face the criminal justice procedure for some shitty personal behaviour on his part.

From what I recall about the situation from past threads is that Sweden cannot send extradition requests in advance. Most criminal courts don't go out of their way to break the rules as that costs and they have to justify the budget.

Also why spend extra time and money on something when they could just wait until he was under British custody? Where is the urgency when normal procedures would suffice?

The Americans went out of their way because this is a political matter to them and their budget was approved.